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6.2 Construct and Interpret Binomial Distributions Page 388 What is a probability distribution? What is a binomial distribution? What is a binomial experiment? Probability Distribution • A probability distribution is a function that gives the probability of each possible value of a random variable. The sum of all the probabilities in a probability distribution must equal one. Probability Distribution for Rolling a Die x 1 2 3 4 5 6 P(x) 1 6 1 6 1 6 1 6 1 6 1 6 Let X be a random variable that represents the sum when two six-sided dice are rolled. Make a table and a histogram showing the probability distribution for X. SOLUTION The possible values of X are the integers from 2 to 12. The table shows how many outcomes of rolling two dice produce each value of X. Divide the number of outcomes for X by 36 to find P(X). a. What is the most likely sum when rolling two six-sided dice? SOLUTION a. The most likely sum when rolling two six-sided dice is the value of X for which P(X) is greatest. This probability is greatest for X = 7. So, the most likely sum when rolling the two dice is 7. b. What is the probability that the sum of the two dice is at least 10? b. The probability that the sum of the two dice is at least 10 is: P(X > 10 ) = P(X = 10) + P(X = 11) + P(X = 12) = 3 1 2 + + 36 36 36 = 6 36 = 1 6 0.167 6.2 Assignment day 1 Page 391, 1-9 6.2 Construct and Interpret Binomial Distributions day 2 Page 388 Binomial Distribution A binomial distribution shows the probabilities of the outcomes of a binomial experiment. Binomial Experiments: There are independent trials. Each trial has only two possible outcomes: success and failure. The probability of success is the same for each trial. This probability is denoted by . The probability of failure is given by 1 − . For a binomial experiment, the probability of exactly successes in n trials is: P( successes)= (1 − )− Sports Surveys According to a survey, about 41% of U.S. households have a soccer ball. Suppose you ask 6 randomly chosen U.S. households whether they have a soccer ball. Draw a histogram of the binomial distribution for your survey. SOLUTION The probability that a randomly selected household has a soccer ball is p = 0.41. Because you survey 6 households, n = 6. P(k = 0) = 6C0(0.41)0(0.59)6 0.042 P(k = 1) = 6C1(0.41)1(0.59)5 0.176 P(k = 2) = 6C2(0.41)2(0.59)4 0.306 P(k = 3) = 6C3(0.41)3(0.59)3 0.283 P(k = 4) = 6C3(0.41)4(0.59)2 0.148 P(k = 5) = 6C5(0.41)5(0.59)1 0.041 P(k = 6) = 6C6(0.41)6(0.59)0 0.005 A histogram of the distribution is shown. a. What is the most likely outcome of the survey? SOLUTION a. The most likely outcome of the survey is the value of k for which P(k) is greatest. This probability is greatest for k = 2. So, the most likely outcome is that 2 of the 6 households have a soccer ball. b. What is the probability that at most 2 households have a soccer ball? SOLUTION b. The probability that at most 2 households have a soccer ball is: P(k < 2) = P(k = 2) + P(k = 1) + P(k = 0) 0.306 + 0.176 + 0.042 ANSWER 0.524 So, the probability is about 52%. Classifying Distributions Probability Distributions can be classified in two ways: Symmetric if a vertical line can be drawn that divides the histogram into two parts that are mirror images. Skewed if the distribution is not symmetric. Describe the shape of the binomial distribution that shows the probability of exactly k successes in 8 trials if (a) p = 0.5 SOLUTION a. Symmetric; the left half is a mirror image of the right half. Describe the shape of the binomial distribution that shows the probability of exactly successes in 8 trials if (b) = 0.9. SOLUTION b. Skewed; the distribution is not symmetric about any vertical line. A binomial experiment consists of 5 trials with probability p of success on each trial. Describe the shape of the binomial distribution that shows the probability of exactly k successes if (a) p = 0.4 and (b) p = 0.5 ANSWER a. The distribution is skewed since it is not symmetric about a vertical line. b. The distribution is symmetric since it is symmetric about a vertical line. How can you figure this out without a histogram? As p gets farther from 0.5, its shape becomes more skewed. What is a binomial distribution? • A probability distribution is a function that gives the probabilities of each possible outcome for a random variable. • The sum of all probabilities in a probability distribution must equal 1. • For a binomial experiment, the probability of exactly successes in trials can be calculated from (1 − )− where n is the number of independent trials and is the probability of success. • A binomial distribution is a probability distribution that shows the probabilities of the outcomes of a binomial experiment. To construct a binomial distribution, list each of the possible outcomes. Then calculate the probability of each of the outcomes. 6.2 Assignment day 2 Page 391, 13-15, 21-23, 30-38, 43-45